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On the Chromatic Thresholds of Hypergraphs

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  06 March 2015

JÓZSEF BALOGH ,
JANE BUTTERFIELD ,
PING HU ,
JOHN LENZ  and
DHRUV MUBAYI
JÓZSEF BALOGH
Affiliation:
Department of Mathematics, University of Illinois, Urbana-Champaign, IL 61801, USA (e-mail: jobal@math.uiuc.edu, pinghu1@math.uiuc.edu)
JANE BUTTERFIELD
Affiliation:
University of Victoria, Victoria BC V8P 5C2, Canada (e-mail: jvbutter@uvic.ca)
PING HU
Affiliation:
Department of Mathematics, University of Illinois, Urbana-Champaign, IL 61801, USA (e-mail: jobal@math.uiuc.edu, pinghu1@math.uiuc.edu)
JOHN LENZ
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, IL 60607, USA (e-mail: lenz@math.uic.edu, mubayi@math.uic.edu)
DHRUV MUBAYI
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, IL 60607, USA (e-mail: lenz@math.uic.edu, mubayi@math.uic.edu)
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Abstract

Let $\mathcal{F}$ be a family of r-uniform hypergraphs. The chromatic threshold of $\mathcal{F}$ is the infimum of all non-negative reals c such that the subfamily of $\mathcal{F}$ comprising hypergraphs H with minimum degree at least $c \binom{| V(H) |}{r-1}$ has bounded chromatic number. This parameter has a long history for graphs (r = 2), and in this paper we begin its systematic study for hypergraphs.

Łuczak and Thomassé recently proved that the chromatic threshold of the so-called near bipartite graphs is zero, and our main contribution is to generalize this result to r-uniform hypergraphs. For this class of hypergraphs, we also show that the exact Turán number is achieved uniquely by the complete (r + 1)-partite hypergraph with nearly equal part sizes. This is one of very few infinite families of non-degenerate hypergraphs whose Turán number is determined exactly. In an attempt to generalize Thomassen's result that the chromatic threshold of triangle-free graphs is 1/3, we prove bounds for the chromatic threshold of the family of 3-uniform hypergraphs not containing {abc, abd, cde}, the so-called generalized triangle.

In order to prove upper bounds we introduce the concept of fibre bundles, which can be thought of as a hypergraph analogue of directed graphs. This leads to the notion of fibre bundle dimension, a structural property of fibre bundles that is based on the idea of Vapnik–Chervonenkis dimension in hypergraphs. Our lower bounds follow from explicit constructions, many of which use a hypergraph analogue of the Kneser graph. Using methods from extremal set theory, we prove that these Kneser hypergraphs have unbounded chromatic number. This generalizes a result of Szemerédi for graphs and might be of independent interest. Many open problems remain.

MSC classification

Primary: 05C35: Extremal problems
Secondary: 05C65: Hypergraphs 05C15: Coloring of graphs and hypergraphs 05D40: Probabilistic methods
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 2 , March 2016 , pp. 172 - 212
Copyright
Copyright © Cambridge University Press 2015 

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